Problem: Determine how many solutions exist for the system of equations. ${x+y = 10}$ ${y = 3-x}$
Convert both equations to slope-intercept form: ${x+y = 10}$ $x{-x} + y = 10{-x}$ $y = 10-x$ ${y = -x+10}$ ${y = 3-x}$ ${y = -x+3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -x+10}$ ${y = -x+3}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.